# Reasoning About Agent Types and the Hardest Logic Puzzle Ever

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## Abstract

In this paper, we first propose a simple formal language to specify types of agents in terms of necessary conditions for their announcements. Based on this language, types of agents are treated as ‘first-class citizens’ and studied extensively in various dynamic epistemic frameworks which are suitable for reasoning about knowledge and agent types via announcements and questions. To demonstrate our approach, we discuss various versions of Smullyan’s *Knights and Knaves* puzzles, including the *Hardest Logic Puzzle Ever* (HLPE) proposed by Boolos (in Harv Rev Philos 6:62–65, 1996). In particular, we formalize HLPE and verify a classic solution to it. Moreover, we propose a spectrum of new puzzles based on HLPE by considering subjective (knowledge-based) agent types and relaxing the implicit epistemic assumptions in the original puzzle. The new puzzles are harder than the previously proposed ones in the literature, in the sense that they require deeper epistemic reasoning. Surprisingly, we also show that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. Our formalism paves the way for studying these new puzzles using automatic model checking techniques.

## Keywords

Agent types Public announcement logic Questioning strategy Knight and Knaves The hardest logic puzzle ever## Notes

### Acknowledgments

The authors would like to thank Hans van Ditmarsch and Johan van Benthem for their detailed comments on earlier versions of this paper, and thank Gregory Wheeler for pointing out the literature on the HLPE, which helped to shape the development of this work. We are also grateful to two anonymous referees of this journal for their very valuable comments. Both authors are partially supported by the Major Program of National Social Science Foundation of China (NO.11&ZD088). Yanjing Wang is also supported by the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities (No.12JJD720011).

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